Problem: Guillermo is a professional deep water free diver. His altitude (in meters relative to sea level), $x$ seconds after diving, is modeled by $g(x)=\dfrac{1}{20}x(x-100)$ What is the lowest altitude Guillermo will reach?
Solution: Guillermo's altitude is modeled by a quadratic function, whose graph is a parabola. The lowest altitude is reached at the vertex. So in order to find the lowest altitude, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $g(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} g(x)&=0 \\\\ \dfrac{1}{20}x(x-100)&=0 \\\\ \swarrow &\searrow \\\\ \dfrac{1}{20}x=0\text{ or }&x-100=0 \\\\ x={0}\text{ or }&x={100} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({100})}{2}=\dfrac{100}{2}={50}$ The vertex's $x$ -coordinate is ${50}$. Now let's find $g({50})$ : $\begin{aligned} g({50})&=\dfrac{1}{20}({50})({50}-100) \\\\ &=\dfrac{1}{20}(50)(-50) \\\\ &=-125 \end{aligned}$ In conclusion, the lowest altitude Guillermo will reach is $-125$ meters relative to sea level.